Question

How far would a submarine periscope have to be above the water for lookout to locate a ship 5 miles away?

Answer 1

I assume you have a formula for the distance to the horizon for a given height ...

Answer 2

I don't know what formula you have been given, but a more or less

general one states

d = √(2h R), where R is the radius , h is the height above the water, and
d is the distance we can see, as long we use the same units throughout.
The radius of the earth is appr 4000 miles
so we want
3 = √(8000h)
9 = 8000h
h = 9/8000 miles
but we probably want this in feet, so multiply by 5280

height = 9/8000*5280 = appr 5.94 ft

Answer 3

oops,

should have been 5 miles instead of 3
I will let you make the necessary changes.
Let me know what your answer is.

Answer 4

[AI-generated answer — Explain Bot]

To determine how far a submarine periscope would need to be above the water for a lookout to locate a ship 5 miles away, we can use basic trigonometry.

The periscope acts as a viewing instrument that allows observers to see objects above the water's surface while the submarine remains submerged. We can consider the periscope as forming a right triangle with the observer's line of sight, the distance from the periscope to the water's surface, and the distance from the periscope to the ship.

Let's assume the observer's eye level is a constant height above the water, represented as "h". The distance from the periscope to the ship is the hypotenuse, which is 5 miles (or approximately 8 kilometers).

By applying the Pythagorean theorem, we can find the distance from the periscope to the water's surface, which can give us the elevation required for the periscope.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b): c^2 = a^2 + b^2.

In our case:

c^2 = 5^2 + h^2.

Simplifying the equation:

h^2 = c^2 - 5^2,
h^2 = 8^2 - 5^2,
h^2 = 64 - 25,
h^2 = 39.

Therefore, h ≈ √39 ≈ 6.24.

The observer's eye level (h) above the water's surface should be approximately 6.24 units (feet, meters, etc.). This means the periscope should extend 6.24 units above the water for the lookout to locate a ship 5 miles away.